Finite Group Databasebeta

← #277013

Subgroups — #277013

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33201C211yesmaximal, characteristic
2#28616C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic
4C4×#474C211yesmaximal, characteristic
5C2×#1683C211yesmaximal, characteristic
6C2×#1669C211yesmaximal, characteristic
7#10593C211yesmaximal, characteristic, complemented
8#10592C211yesmaximal, characteristic, complemented
9#10541C211yesmaximal, characteristic, complemented
10#10538C211yesmaximal, characteristic, complemented
11#9693C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9219C211yesmaximal, characteristic
14#9215C211yesmaximal, characteristic
15#8591C211yesmaximal, characteristic, complemented
16C2×#494C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#474C2211yescharacteristic
20C2×#474C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2862C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#2056—21no—
27#2056—21no—
28#1990C2211yescharacteristic
29#1988C2211yescharacteristic
30#1988C2211yescharacteristic
31#1987C2211yescharacteristic
32#1956C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1918—21no—
39#1878—21no—
40#1878—21no—
41#1862—21no—
42#1850—21no—
43#1850—21no—
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1804—21no—
47#1804—21no—
48#1802C2211yescharacteristic
49#1797—21no—
50C4×#107—21no—
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×#102—21no—
54C2×#336—21no—
55#1683—24no—
56#1669—24no—
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×Q16—21no—
70C22×Q16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4×Q8—21no—
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42—21no—
81#494—22no—
82#494—41no—
83#487—22no—
84#487—41no—
85#476—22no—
86#476—22no—
87#474—22no—
88#474—22no—
89#474—22no—
90#474D414yes—
91C8×D4—22no—
92C8×D4—22no—
93C8×D4—41no—
94C8×D4—41no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#107—21no—
98C2×#106—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391—21no—
129#389—21no—
130#389—21no—
131#384C2311yescharacteristic
132#382—21no—
133#382—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C43C2311yescharacteristic
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—22no—
161#335—22no—
162#333—22no—
163#333—41no—
164#332—22no—
165#332—22no—
166#332—22no—
167#332—41no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C2×Q16—42no—
176C2×Q16—42no—
177C2×Q16—42no—
178C2×Q16—42no—
179C2×Q16—81no—
180C2×Q16—81no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C2×D412yes—
209C4×Q8—81no—
210C4×Q8—81no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—41no—
216C4×D4Q1614yescomplemented
217C4×D4C2×D412yes—
218C4×D4#4112yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42—81no—
240C2×C42—81no—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248#107—24no—
249#107—42no—
250#107—42no—
251#106—42no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—41no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—81no—
268C4⋊C8—81no—
269C4⋊C8C2×D412yes—
270#103—42no—
271#103—42no—
272#103—81no—
273#103—81no—
274#102—24no—
275#102—42no—
276#102—42no—
277#102—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—42no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—41no—
289#95—41no—
290#95—41no—
291#95—41no—
292#95—41no—
293#95—41no—
294C24—21no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—81no—
310C2×Q8—81no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4#12112yes—
315C2×D4C2×Q1612yes—
316C2×D4#13712yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343Q16—84no—
344Q16—84no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—84no—
375C2×C8—84no—
376C2×C8#13512yes—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C4⋊C4#12112yes—
400C4⋊C4C2×Q1612yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—81no—
416C42—81no—
417C42—82no—
418C42C2×Q1612yes—
419C42#13512yes—
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C422−1+411yescharacteristic
424C23—22no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C2×#12111yescharacteristic
432Q8—42no—
433Q8—42no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438Q8—81no—
439Q8—81no—
440Q8—161no—
441Q8—161no—
442D4—22no—
443D4—22no—
444D4—22no—
445D4#46214yes—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4#46712yes—
486C2×C4#46712yes—
487C2×C4C2×#12111yescharacteristic
488C2×C4#53711yescharacteristic
489C2×C4D4211yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4#55311yescharacteristic
493C2×C4C22×Q1611yescharacteristic
494C2×C4C2×#13511yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57511yescharacteristic
498C2×C4#57511yescharacteristic
499C2×C4#57611yescharacteristic
500C2×C4#57611yescharacteristic
501C8—22no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—42no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8—81no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22#241611yescharacteristic
519C22#303511yescharacteristic
520C22#304111yescharacteristic
521C22C2×#46711yescharacteristic
522C22#305611yescharacteristic
523C22#306711yescharacteristic
524C22#306911yescharacteristic
525C4—22no—
526C4—22no—
527C4—21no—
528C4—21no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—41no—
533C4—81no—
534C4—82no—
535C4—161no—
536C4—161no—
537C4—161no—
538C4C2×#46211yescharacteristic
539C4#303511yescharacteristic
540C4#324811yescharacteristic
541C4#324911yescharacteristic
542C4#327311yescharacteristic
543C4#328011yescharacteristic
544C4#336111yescharacteristic
545C4#337011yescharacteristic
546C2—44no—
547C2#1912611yescharacteristic
548C2#1918111yescharacteristic
549C2#1938711yescharacteristic
550C2#1946011yescharacteristic
551C2#2473511yescharacteristic
552C2#2475211yescharacteristic
553C2#2477511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #28616C2semidirect
7. #10593C2semidirect
8. #10592C2semidirect
9. #10541C2semidirect
10. #10538C2semidirect
11. #9693C2semidirect
12. #9601C2semidirect
15. #8591C2semidirect
216. C4×D4Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332012
#286162
#226962
C4×#4742
C2×#16832
C2×#16692
#105932
#105922
#105412
#105382
#96932
#96012
#92192
#92152
#85912